Example of a paracompact space that is not metrizable I'm looking for an example of a space which is paracompact but not metrizable. The definition of paracompactness that I'm working with is that $(X,\tau)$ is paracompact if it is Hausdorff ($T_{2}$) and for every open cover there exists a locally finite open refinement.
I'd also like to know how well paracompactness is preserved in products. Thanks in advance.
 A: The Sorgenfrey line is a classical example (besides the compact examples mentioned in the thread from the comments), also because its square (the Sorgenfrey plane) is not even normal, let alone paracompact, which shows that products of even 2 relatively nice paracompact spaces can fail to be paracompact.
As a positive result, the product of a paracompact and a compact Hausdorff space is paracompact.
A: $\pi$-Base is an online encyclopedia of topological spaces inspired by Steen and Seebach's Counterexamples in Topology. It lists the following paracompact, Hausdorff spaces that are not metrizable. You can learn more about any of them by visiting the search result.
Alexandroff Square
Appert Space
Arens-Fort Space
Closed Ordinal Space $[0,\Omega]$
Concentric Circles
Fortissimo Space
Helly Space
$I^I$
Lexicographic Ordering on the Unit Square
Radial Interval Topology
Right Half-Open Interval Topology
Single Ultrafilter Topology
Stone-Cech Compactification of the Integers
The Extended Long Line
Tychonoff Plank
Uncountable Fort Space
